Calculus Answers

Find equations of all lines having slope -1 that are tangent to the curve 𝑦=1/𝑥−1

Evaluate the Integration of Rational Functions by Partial Fractions:

1. ∫ (x² +x-1/x³-x) dx

2. ∫ (x² -x+1/(x+1)³) dx

3. ∫ (x²+4x+10/x³ +2x² +5x) dx

4. ∫ (x²/(x-1)(x⁴ +8x² +16)) dx

Evaluate the the Integration by Trigonomentric Substitution:

1. ∫ √x² -1 dx/x²

2. ∫ dx/√6x+x²

3. ∫ dx/x√x⁴ -4

4. ∫ x² dx/√9-4x²

11. (Section 7.9 and Chapter 8) Consider the 3–dimensional vector field F defined by F (x, y, z) = ￾ 12x 2 y 2 + 2z 2 + 1, 8x 3 y − 3z, 4xz − 3y − 3
 (a) Write down the Jacobian matrix JF (x, y, z). (2) (b) Determine the divergence div F(x, y, z). (2) (c) Determine curl F(x, y, z). (2) (d) Give reasons why F has a potential function. (Refer to the relevant definitions and theorems in the study guide.) (2) (e) Find a potential function of F, using the method of Example 7.9.1. Note, however that that example concerns a 2-dimensional vector field, so you will have to adapt the method to be suitable for a 3-dimensional vector field. Pay special attention to the notation that you use for derivatives of functions of more than one variable.

10. (Sections 11.1 - 11.3, 7.5, 9.3) (a) State the Implicit Function Theorem for an equation in the three variables x, y and z. (2) (b) Consider the equation xyz = 4x 2 + y 2 − z 2 . Use the Implicit Function Theorem to show that the given equation has a smooth unique local solution of the form z = g(x, y) about the point (2, 0, 4). Then find the local linearization of g about the point (2, 0). Hints: • Use the method of Example 11.2.6, but take into account that you are dealing with an equation in three variables here and that g in this case is a function of two variables. • Before you apply the Implicit Function Theorem, you should show that all the necessary conditions are satisfied. • Study Remark 11.3.3(1) and Remark 9.3.6(2).

9. (Sections 2.12, 3.2, 7.5, 7.8) Consider the R 2 − R function f defined by f (x, y) = xy and let C be the contour curve of f at level 4. (a) Find a Cartesian equation for the tangent L to C at (x, y) = (1, 4). (4) (b) Sketch the contour curve C together with the line L in R 2 . (3) (c) Find an equation for the tangent plane V to the graph of f at (x, y) = (1, 4). (4) Hints: • Study Definition 3.2.5 and Remarks 3.2.6 again and note that the contour curves of f lie in the XY-plane. Also study the more general Definition 7.8.4 and read Remarks 7.8.5. • Use Theorem 7.8.1 to find a vector n which is perpendicular to L and then use Definition 2.12.1 to find an equation for L, OR use Definition 7.8.6 directly. Note that in the case of n = 2, the formula in Definition 7.8.6 gives a Cartesian equation for the tangent to a contour curve. • Study Definition 7.5.4 and use the formula (7.2) to find an equation for V OR define a function g for which the graph of f is a contour surface

 8. (Sections 2.12, 7.8) Consider the surface S =  (x, y, z) ∈ R 3 | x 2 + y 2 + z 2 = 9 . (a) Define an R 3 − R function f such that S is the contour surface of f at level 9. (1) (b) Use Definition 7.8.6 to find an equation for the plane V that is tangent to S at the point (x, y, z) = (2, 1, 2). (4) (c) Sketch the surface S in R 3 , together with a section of the plane V to illustrate that V is tangent to S at the point (2, 1, 2)

 7. (Sections 9.1, 9.2) Consider the R − R function f defined by f (x) = e x ln x. Determine the third order Taylor polynomial T3 (x) of f about the point 1

6. (Sections 1.3, 3.3, 7.6) Consider the R 2 − R function f defined by f(x, y) = ln xy and the R − R 2 function r defined by r(t) = ￾ t 2 , et
. Determine the value of (f ◦ r) 0 (1) by using the General Chain Rule (Theorem 7.6.1). Hints: • Determine (f ◦ r) 0 (1) directly from the formula given in Theorem 7.6.1. It is unnecessary to first determine (f ◦ r) 0 (t). • Check your answer by finding the composite function f ◦r, then finding the derivative (f ◦ r) 0 (t) and finally putting t = 1

5. (Sections 6.1, 6.3) Consider the R − R 2 function r defined by r (t) = ￾ t, t2
; t ∈ [−3, 3] . (a) Determine the vector derivative r 0 (1) by using Definition 6.1.1(b) Sketch the curve r together with the vector r 0 (1), in order to illustrate the geometric meaning of the vector derivative. Note: The curve r is the image of r, so it consists of all points (x, y) = (t, t2 ); t ∈ [−3, 3]